We optimize the possible geometrical cluster structures and predicte relative stability of (HMgN3)n(n=15) by using the hybrid density functional theory (B3LYP) with 6-311G* basis sets. And the most stable isomers of (HMgN3)n(n=15) clusters, the bond properties, charge distributions, vibrational properties, and stability are analyzed theoretically. The calculated results show that the most stable HMgN3 has a linear structure, the (HMgN)n(n=2,5) clusters have the most stable structures in which an N atom in a sub-system and metal atom in another sub-system constitute an MgNMg structure. And the most stable structures of (HMgN3)n(n=3,4) clusters are the chain structures in which the nitrogen cardinal extremity position N atom and the Mg atom form a ring structure; the metal Mg atoms in the most stable structure show charge positivity, and H atom show charge negativity. The middle N atoms of azido show charge positivity, the N atoms on both sides of azido show charge negativity; what's more, the N atoms influenced by Mg atoms directly show a more charge negativity. MgN bond and MgH bond are the typical ionic bond; the bond between N atoms in azido is the covalent bond. The infrared spectra of the most optimized (HMgN3)n(n=15) clusters have three vibrational sections, the strongest vibrational peak lies in 22582347 cm-1, and the vibrational mode is anti-symmetric stretching vibration of NN bonds in azido. Analysis of stability shows that (HMgN3)3 clusters are more stable than other clusters.